#139184

order	512
nilpotency class	4
isoclinism	isoclinic class of #139144
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1893
commutator	C₄×C₈	C₂²×C₄
Frattini, cosocle	C₂×C₄×C₈	C₂³
prime radical	C₂×#41	D₈
LCS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₈×#41 ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

345 conjugacy classes of subgroups; only the 98 normal subgroups are shown.

normal subgroups only

This group has 345 conjugacy classes of subgroups — too many to draw. Only its 98 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 3 generators on 32 points:

(1,2,5,9,6,10,4,8)(3,11,13,7,14,15,12,16)(18,22)(19,23)(20,26)(24,32)(25,30)(27,31)(28,29);(1,3)(2,7)(4,13)(5,12)(6,14)(8,15)(9,11)(10,16)(17,18,23,29,20,24,27,30,21,25,31,32,26,28,19,22);(1,3)(2,7)(4,12)(5,13)(6,14)(8,11)(9,15)(10,16)

Polycyclic presentation:

2^9|a^2=f,b^2=dfg,d^2=ghi,e^2=h,f^2=h,g^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=eh,g^a=gi,d^b=dh,f^b=fh,e^c=eh

Conjugacy classes & character table

83 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 46380);